skip to main content

Title: Massively Parallel Algorithms for Distance Approximation and Spanners.
Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms---usually sublogarithmic-time and often poly(łogłog n)-time, or even faster---for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present poly(łog k) ε poly(łogłog n) round MPC algorithms for computing O(k^1+o(1) )-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an O(łog^2łog n)-round algorithm for O(łog^1+o(1) n) approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for more » approximating APSP in weighted graphs in the Congested Clique model. « less
Authors:
; ; ; ;
Award ID(s):
2006664
Publication Date:
NSF-PAR ID:
10339759
Journal Name:
33rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2021)
Sponsoring Org:
National Science Foundation
More Like this
  1. We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with ˜O(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the ˜O (plog Δ)-round algorithm of Ghaffari [PODC’17]. • OurO(log logn)-round (1+ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log2 logn)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC’18] and O(log logn)-round (1 + ε)- approximation algorithm of Assadi et al. [arXiv’17]. • Our O(log logn)-round (2+ε)-approximate minimum vertex cover algorithm improves on an O(log logn)-round O(1)- approximation of Assadi et al. [arXiv’17].
  2. Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower boundsmore »for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.« less
  3. The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)-approximation in Õ(n) time or 3.5-approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approxima- tions are inevitable in the near-linear time regime. To complement the lower bound, we provide a 3.3-approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Building on our construction we prove near-quadratic lower bounds under SETHmore »for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGET model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.« less
  4. We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost k-cycle free graphs, for any constant k≥ 4. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4- or 5-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve super-constant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3-SUM or APSP conjectures. In particular, we prove that k-approximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4-Cycle problem: An infamous open question in fine-grained complexity is to establish any surprising consequences from amore »subquadratic or even linear-time algorithm for detecting a 4-cycle in a graph. This is arguably one of the simplest problems without a near-linear time algorithm nor a conditional lower bound. We prove that Ω(m1.1194) time is needed for k-cycle detection for all k≥ 4, unless we can detect a triangle in √n-degree graphs in O(n2−δ) time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms.« less
  5. Counting and uniformly sampling motifs in a graph are fundamental algorithmic tasks with numerous applications across multiple fields. Since these problems are computationally expensive, recent efforts have focused on devising sublinear-time algorithms for these problems. We consider the model where the algorithm gets a constant size motif H and query access to a graph G, where the allowed queries are degree, neighbor, and pair queries, as well as uniform edge sample queries. In the sampling task, the algorithm is required to output a uniformly distributed copy of H in G (if one exists), and in the counting task it is required to output a good estimate to the number of copies of H in G. Previous algorithms for the uniform sampling task were based on a decomposition of H into a collection of odd cycles and stars, denoted D∗(H) = {Ok1 , ...,Okq , Sp1 , ..., Spℓ19 }. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, for any motif H whose decomposition contains at least twomore »components or at least one star, is always preferable. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. We further show how to use our sampling algorithm to get an approximate counting algorithm, with essentially the same complexity. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif HD 30 with decomposition D, and a family of graphs G, so that in order to output a uniform copy of H in a uniformly chosen graph in G, the number of required queries matches our upper bound. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.« less